Ellipsoid

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Ellipsoid_3d.jpg
3D rendering of an ellipsoid
 Contents

Definition

In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid in an x-y-z Cartesian coordinate system is

[itex]

{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1 [itex] where a, b and c are fixed positive real numbers determining the shape of the ellipsoid. If two of those numbers are equal, the ellipsoid is a spheroid; if all three are equal, we have a sphere.

If we assume a ≥ b ≥ c, then when:

• a ≠ b ≠ c we have a scalene ellipsoid
• c = 0 we have a flat ellipsoid (two ellipses back to back)
• b = c we have a prolate spheroid (cigar-shaped)
• a = b we have an oblate spheroid (pill-shaped)
• a = b = c we have a sphere, as stated earlier

Volume

The volume of an ellipsoid is given by:

[itex]\frac{4}{3} \pi abc[itex]

Surface area

The surface area of an ellipsoid is given by:

[itex]2 \pi \left( c^2 + \frac{bc^2}{\sqrt{a^2-c^2}} F(\theta, m) + b\sqrt{a^2-c^2} E(\theta, m) \right)[itex]

where

[itex]m = \frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}[itex]
[itex]\theta = \arcsin{\left( e \right)}[itex]
[itex]e = \sqrt{1 - \frac{c^2}{a^2}}[itex]

and [itex]F(\theta, m)[itex] and [itex]E(\theta, m)[itex] are the incomplete elliptic integrals of the first and second kind.

Exact formulae are:

If flat: [itex]= 2 \pi \left( ab \right)[itex]
If prolate: [itex]= 2 \pi \left( c^2 + ac \frac{\arcsin{\left( e \right)}}{e} \right)[itex]
If oblate: [itex]= 2 \pi \left( a^2 + c^2 \frac{\operatorname{arctanh}{\left( e \right)}}{e} \right)[itex]

Approximate formula is:

If scalene: [itex]\approx 4 \pi \left( \frac{ a^p b^p + a^p c^p + b^p c^p }{3} \right)^{1/p}[itex]

Where p ≈ 1.6075 yields a relative error of at most 1.061% (Knud Thomsen's formula); a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178% (David W. Cantrell's formula).

Linear transformations

If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the spectral theorem.

The intersection of an ellipsoid with a plane is empty, a single point or an ellipse.

One can also define ellipsoids in higher dimensions, as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.

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